Disjoint empty disks supported by a point set

Abstract

For a planar point-set $P$, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in $P$ lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and Urabe recently conjectured that $D(n)=\lceil n/2\rceil$. Here we show that $D(n)\ge n/2+n/236-O(\sqrt{n})$ and thereby disprove this conjecture.

19 pages, 8 figures; minor update; updated references